Equivariant Heegaard genus of reducible 3-manifolds

TL;DR

This paper investigates the equivariant Heegaard genus of reducible 3-manifolds with finite group actions, revealing its complex behavior under connected sums and establishing bounds using orbifold thin position theory.

Contribution

It introduces a new perspective on the equivariant Heegaard genus for reducible 3-manifolds and develops bounds using orbifold thin position theory.

Findings

Equivariant Heegaard genus can be super-additive, additive, or sub-additive under equivariant connected sum.

Sharp bounds on the equivariant Heegaard genus are established for reducible manifolds.

The paper applies thin position theory for 3-orbifolds to analyze genus behavior.

Abstract

The equivariant Heegaard genus of a 3-manifold $M$ with the action of a finite group $G$ of diffeomorphisms is the smallest genus of an equivariant Heegaard splitting for $M$. Although a Heegaard splitting of a reducible manifold is reducible and although if $M$ is reducible, there is an equivariant essential sphere, we show that equivariant Heegaard genus may be super-additive, additive, or sub-additive under equivariant connected sum. Using a thin position theory for 3-dimensional orbifolds, we establish sharp bounds on the equivariant Heegaard genus of reducible manifolds, similar to those known for tunnel number.